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There are 32 students who can opt for classes maths, chemistry and physics. Maths are taken by 13 students, physics by 15 and chemistry by 14. 3 students are taking all 3 classes, 3 are taking ONLY physics and chemistry, 4 are taking ONLY maths and physics and 5 of them are not taking any classes. How many are taking only maths, only physics and only chemistry?

If I draw out the diagram, I end up with 6 people who are studying only maths, 5 only physics and 8 only chemistry. But that doesn't add up to 32 students. I'm confused about the terms "taking ONLY maths and physics,..." since it tells us that these students are completely separate from the ones who are taking all 3, therefore it seems correct to approach the problem the way I did.

Is the problem posed incorrectly, or am I missing something?

ryang
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Mixoftwo
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  • I get 2 taking maths and chemistry but not physics. Seems to me you have not subtracted the off maths-only and off the chemistry only. Still, I can see how the choice of language could lead to confusion. – Will Jagy Apr 01 '23 at 22:20
  • The problem becomes easier to solve if you use a truth table rather than a Venn diagram. – user2661923 Apr 02 '23 at 17:34

1 Answers1

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It looks relatively straightforward to me. I suspect you forgot the 5 students who are not taking any of these three courses!

Draw the standard three overlapping circles labeled "C" for chemistry, "P" for physics, and "M" for mathematics. "3 students are taking all 3 classes" so write 3 in the small center section where all three circles overlap. "3 are taking ONLY physics and chemistry". So write 3 where only P and C overlap. "4 are taking ONLY maths and physics" so write 4 where only M and P overlap. "5 of them are not taking any classes" so write 5 outside all three circles.

The reason this only "relatively" straight forward is that we are not told how many are taking "only math and chemistry" so write "x" where M and C intersect.

We have 4+ 3+ 3= 10 taking physics and 1 or 2 other courses. Since "15 students are taking physics",15- 10= 5 are taking only physics.

We have 4+ 3+ x= 7+ x taking math and 1 or 2 other courses. Since "13 students are taking math" 13- (7+ x)= 6- x are taking only math.

We have 3+ 3+ x= 6+ x taking math and 1 or 2 other courses. Since "14 students are taking chemistry" 14- (6+ x)= 8- x are taking only chemistry.

Now that we have the "overlaps" sorted out, add them up. We have 5+ 3+ 3+ 4= 15, 6- x+ 8- x= 14- 2x, and DON'T FORGET THE 5 WHO ARE NOT TAKING any of these courses! 15+ 14- 2x+ 5= 34- 2x= 32 so 2x= 34- 32= 2 and x= 1. There are 6- 1= 5 students taking only math and 8- 1= 7 students taking only chemistry,

George Ivey
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  • I completely disregarded the students taking chemistry and math. I just assumed there were none haha. Thanks for your answer, very helpful. – Mixoftwo Apr 01 '23 at 23:57